Thermo-physical properties of water and nanoparticles [39].

## 1. Introduction

The natural convection flow over a surface embedded in saturated porous media is encountered in many engineering problems such as the design of pebble-bed nuclear reactors, ceramic processing, crude oil drilling, geothermal energy conversion, use of fibrous material in the thermal insulation of buildings, catalytic reactors and compact heat exchangers, heat transfer from storage of agricultural products which generate heat as a result of metabolism, petroleum reservoirs, storage of nuclear wastes, etc.

The derivation of the empirical equations which govern the flow and heat transfer in a porous medium has been discussed in [1-5]. The natural convection on vertical surfaces in porous media has been studied used Darcy’s law by a number of authors [6–20]. Boundary layer analysis of natural convection over a cone has been investigated by Yih [21-24]. Murthy and Singh [25] obtained the similarity solution for non-Darcy mixed convection about an isothermal vertical cone with fixed apex half angle, pointing downwards in a fluid saturated porous medium with uniform free stream velocity, but a semi-similar solution of an unsteady mixed convection flow over a rotating cone in a rotating viscous fluid has been obtained Roy and Anilkumar [26]. The laminar steady nonsimilar natural convection flow of gases over an isothermal vertical cone has been investigated by Takhar et al. [27]. The development of unsteady mixed convection flow of an incompressible laminar viscous fluid over a vertical cone has been investigated by Singh and Roy [28] when the fluid in the external stream is set into motion impulsively, and at the same time the surface temperature is suddenly changed from its ambient temperature. An analysis has been carried out by Kumari and Nath [29] to study the non-Darcy natural convention flow of Newtonian fluids on a vertical cone embedded in a saturated porous medium with power-law variation of the wall temperature/concentration or heat/mass flux and suction/injection. Cheng [30-34] focused on the problem of natural convection from a vertical cone in a porous medium with mixed thermal boundary conditions, Soret and Dufour effects and with variable viscosity.

The conventional heat transfer fluids including oil, water and ethylene glycol etc. are poor heat transfer fluids, since the thermal conductivity of these fluids play an important role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface. An innovative technique for improving heat transfer by using ultra fine solid particles in the fluids has been used extensively during the last several years. Choi [35] introduced the term nanofluid refers to these kinds of fluids by suspending nanoparticles in the base fluid. Khanafer et al. [36] investigated the heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. The convective boundary-layer flow over vertical plate, stretching sheet and moving surface studied by numerous studies and in the review papers Buongiorno [37], Daungthongsuk and Wongwises [38], Oztop [39], Nield and Kuznetsov [40,41], Ahmad and Pop [42], Khan and Pop [43], Kuznetsov and Nield [44,45] and Bachok et al. [46].

From literature survey the base aim of this work is to study the free convection boundary-layer flow past a vertical cone embedded in a porous medium filled with a nanofluid, the basic fluid being a non-Newtonian fluid by using similarity transformations. The reduced coupled ordinary differential equations are solved numerically. The effects of the parameters governing the problem are studied and discussed.

## 2. Mathematical formulation of the problem

Consider the problem of natural convection about a downward-pointing vertical cone of half angle

The nanofluid properties are assumed to be constant except for density variations in the buoyancy force term. The thermo physical properties of the nanofluid are given in Table 1 (see Oztop and Abu-Nada [39]). Assuming that the thermal boundary layer is sufficiently thin compared with the local radius, the equations governing the problem of Darcy flow through a homogeneous porous medium saturated with power-law nanofluid near the vertical cone can be written in two-dimensional Cartesian coordinates

Where

Here

The associated boundary conditions of Eqs. (1)-(3) can be written as:

where

Pure water | 997.1 | 4179 | 0.613 | 21 |

Copper (Cu) | 8933 | 385 | 401 | 1.67 |

Silver (Ag) | 10500 | 235 | 429 | 1.89 |

Alumina (Al_{2}O_{3}) | 3970 | 765 | 40 | 0.85 |

Titanium Oxide (TiO_{2}) | 4250 | 686.2 | 8.9538 | 0.9 |

By introducing the following non-dimensional variables:

The continuity equation is automatically satisfied by defining a stream function

where;

Integration the momentum Eq. (2) we have:

Substituting variables (6) into Eqs. (1)–(5) with Eq. (9), we obtain the following system of ordinary differential equations:

along with the boundary conditions:

where primes denote differentiation with respect to

where

## 3. Results and discussion

In this study we have presented similarity reductions for the effect of a nanoparticle volume fraction on the free convection flow of nanofluids over a vertical cone via similarity transformations. The numerical solutions of the resulted similarity reductions are obtained for the original variables which are shown in Eqs. (10) and (11) along with the boundary conditions (12) by using the implicit finite-difference method. The physical quantity of interest here is the Nusselt number _{2}O_{3}) and Titanium oxide (TiO_{2}).

In order to verify the accuracy of the present method, we have compared our results with those of Yih [22] for the rate of heat transfer

Vertical plate | Vertical cone | |||

Yih [22] | Present method | Yih [22] | Present method | |

0.5 | 0.3766 | 0.3768 | 0.6522 | 0.6524 |

0.8 | 0.4237 | 0.4238 | 0.7339 | 0.7340 |

1.0 | 0.4437 | 0.4437 | 0.7686 | 0.7686 |

1.5 | 0.4753 | 0.4752 | 0.8233 | 0.8233 |

2.0 | 0.4938 | 0.4938 | 0.8552 | 0.8552 |

0.05 | 0.7423 | 0.7704 | 0.6604 | 0.6725 |

0.1 | 0.6931 | 0.7330 | 0.5642 | 0.5852 |

0.15 | 0.6301 | 0.6732 | 0.4780 | 0.5057 |

0.2 | 0.5591 | 0.6002 | 0.4006 | 0.4331 |

0.3 | 0.4052 | 0.4357 | 0.2673 | 0.3062 |

Figs. 5 and 6 are presented to show the effect of the volume fraction of nanoparticles Cu and Ag respectively, on temperature distribution. These figures illustrate the streamline for different values of

Fig. 8 shows the variation of the reduced Nusselt number with the nanoparticles volume fraction _{2}-nanoparticles and Al_{2}O_{3}-nanoparticles. Also, the Fig. 8 and Table 3 show that the values of

## 4. Conclusions

The problem of the steady free convection boundary layer flow past a vertical cone embedded in porous medium filled with a non-Newtonian nanofluid has been studied and the special case when the base fluid is water has been considered. The effects of the solid volume fraction _{2}O_{3}) and Titanium oxide (TiO_{2}). It has been shown, as expected, that increasing of the values of the nanoparticles volume fraction lead to an increase of the velocity and the temperature profiles and to an decrease of the Nusselt number for the values of the parameter

### Nomenclature

## References

- 1.
Cheng P. 1978 Heat transfer in geothermal systems. Adv. Heat Transfer14 1 105 - 2.
Nield D. A. Bejan A. 1999 Convection in Porous Media second ed., Springer: New York. - 3.
Vafai K. 2000 Handbook of Porous Media Marcel Dekker: New York. - 4.
Pop I. Ingham D. B. 2001 Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media. Pergamon Press: Oxford. - 5.
Ingham D. B. Pop I. 2002 Transport Phenomena in Porous Media Pergamon Press: Oxford - 6.
Cheng P. Minkowycz W. J. 1977 Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike J. Geophys. Res.82 14 2040 2044 - 7.
Gorla R. S. R. Tornabene R. 1988 Free convection from a vertical plate with nonuniform surface heat flux and embedded in a porous medium Transp. Porous Media3 95 106 - 8.
Bakier AY, Mansour MA, Gorla RSR, Ebiana AB 1997 Nonsimilar solutions for free convection from a vertical plate in porous media Heat Mass Transfer33 145 148 - 9.
Gorla RSR, Mansour MA, Abdel-Gaied SM 1999 Natural convection from a vertical plate in a porous medium using Brinkman’s model Transp. Porous Media36 357 371 - 10.
Mulolani I. Rahman M. 2000 Similarity analysis for natural convection from a vertical plate with distributed wall concentration Int. J. Math. math. Sci.23 5 319 334 - 11.
Jumah RY, Mujumdar AS 2000 Free convection heat and mass transfer of non-Newtonian power law fluids with yield stress from a vertical plate in saturated porous media. Int Comm. Heat Mass Transfer27 485 494 - 12.
Groşan T. Pop I. 2001 Free convection over a vertical flat plate with a variable wall temperature and internal heat generation in a porous medium saturated with a non-Newtonian fluid. Technische Mechanik21 4 313 318 - 13.
Kumaran V. Pop I. 2006 Steady free convection boundary layer over a vertical flat plate embedded in a porous medium filled with water at 4^{o}C. Int. J. Heat Mass Transfer49 3240 3252 - 14.
Magyari E. Pop I. Keller B. 2006 Unsteady free convection along an infinite vertical flat plate embedded in a stably stratified fluid-saturated porous medium Transp. Porous Media62 233 249 - 15.
Cheng C-Y 2006 Natural convection heat and mass transfer of non-Newtonian power law fluids with yield stress in porous media from a vertical plate with variable wall heat and mass fluxes Int. Comm. Heat Mass Transfer33 1156 1164 - 16.
Chamkha A. J. Al-Mudhaf A. F. Pop I. 2006 Effect of heat generation or absorption on thermophoretic free convection boundary layer from a vertical flat plate embedded in a porous medium Int. Comm. Heat Mass Transfer33 1096 1102 - 17.
Magyari E. Pop I. Postelnicu A. 2007 Effect of the source term on steady free convection boundary layer flows over an vertical plate in a porous medium. Part I Transp. Porous Media67 49 67 - 18.
Nield DA, Kuznetsov AV 2008 Natural convection about a vertical plate embedded in a bidisperse porous medium Int. J. Heat Mass Transfer51 1658 1664 - 19.
Mahdy A. Hady F. M. 2009 Effect of thermophoretic particle deposition in non-Newtonian free convection flow over a vertical plate with magnetic field effect J. Non-Newtonian Fluid Mech.161 37 41 - 20.
Ibrahim FS, Hady FM, Abdel-Gaied SM, Eid MR 2010 Influence of chemical reaction on heat and mass transfer of non-Newtonian fluid with yield stress by free convection from vertical surface in porous medium considering Soret effect Appl. Math. Mech.-Engl. Ed.31 6 675 684 - 21.
Yih KA 1997 The effect of uniform lateral mass flux effect on free convection about a vertical cone embedded in a saturated porous medium. Int. Comm. Heat Mass Transfer24 8 1195 1205 - 22.
Yih KA 1998 Uniform lateral mass flux effect on natural convection of non-Newtonian fluids over a cone in porous media Int. Comm. Heat Mass Transfer25 7 959 968 - 23.
Yih KA 1999 Effect of radiation on natural convection about a truncated cone Int. J. Heat Mass Transfer42 4299 4305 - 24.
Yih KA 1999 Coupled heat and mass transfer by free convection over a truncated cone in porous media: VWT/VWC or VHF/VMF Acta Mech.137 83 97 - 25.
Murthy P. V. S. N. Singh P. 2000 Thermal dispersion effects on non-Darcy convection over a cone Compu. Math. Applications40 1433 1444 - 26.
Roy S. Anilkumar D. 2004 Unsteady mixed convection from a rotating cone in a rotating fluid due to the combined effects of thermal and mass diffusion Int. J. Heat Mass Transfer47 1673 1684 - 27.
Takhar H. S. Chamkha A. J. Nath G. 2004 Effect of thermophysical quantities on the natural convection flow of gases over a vertical cone Int. J. Eng. Sci.42 243 256 - 28.
Singh P. J. Roy S. 2007 Unsteady mixed convection flow over a vertical cone due to impulsive motion Int. J. Heat Mass Transfer50 949 959 - 29.
Kumari M. Nath G. 2009 Natural convection from a vertical cone in a porous medium due to the combined effects of heat and mass diffusion with non-uniform wall temperature/concentration or heat/mass flux and suction/injection Int. J. Heat Mass Transfer52 3064 3069 - 30.
Cheng C-Y 2009 Natural convection heat transfer of non-Newtonian fluids in porous media from a vertical cone under mixed thermal boundary conditions Int. Comm. Heat Mass Transfer36 693 697 - 31.
Cheng C-Y 2009 Soret and Dufour effects on natural convection heat and mass transfer from a vertical cone in a porous medium Int. Comm. Heat Mass Transfer36 1020 1024 - 32.
Cheng C-Y 2010 Soret and Dufour effects on heat and mass transfer by natural convection from a vertical truncated cone in a fluid-saturated porous medium with variable wall temperature and concentration Int. Comm. Heat Mass Transfer37 1031 1035 - 33.
Cheng C-Y 2010 Soret and Dufour effects on heat and mass transfer by natural convection from a vertical truncated cone in a fluid-saturated porous medium with variable wall temperature and concentration Int. Comm. Heat Mass Transfer37 1031 1035 - 34.
Cheng C-Y 2010 Nonsimilar boundary layer analysis of double-diffusive convection from a vertical truncated cone in a porous medium with variable viscosity Int. Comm. Heat Mass Transfer37 1031 1035 - 35.
Choi SUS 1995 Enhancing thermal conductivity of fluid with nanoparticles. developments and applications of non-Newtonian flow. ASME FED 231/MD66 99 105 - 36.
Khanafer K. Vafai K. Lightstone M. 2003 Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids Int. J. Heat Mass Transfer46 3639 3653 - 37.
Buongiorno J. 2006 Convective transport in nanofluids ASME J. Heat Transfer128 240 250 - 38.
Daungthongsuk W. Wongwises S. 2007 A critical review of convective heat transfer nanofluids. R en. Sustainable Energy Rev.11 797 817 - 39.
Oztop H. F. Abu-Nada E. 2008 Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids Int. J. Heat Fluid Flow29 1326 1336 - 40.
Nield DA, Kuznetsov AV 2009 The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluids. Int. J. Heat Mass Transfer52 5792 5795 - 41.
Nield DA, Kuznetsov AV 2011 The Cheng-Minkowycz problem for the double-diffusive natural convective boundary-layer flow in a porous medium saturated by a nanofluids. Int. J. Heat Mass Transfer54 374 378 - 42.
Ahmad S. Pop I. 2010 Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids Int. Comm. Heat Mass Transfer37 987 991 - 43.
Khan W. A. Pop I. 2010 Boundary-layer flow of a nanofluid past a stretching sheet Int. J. Heat Mass Transfer53 2477 2483 - 44.
Kuznetsov AV, Nield DA 2010 Natural convective boundary-layer flow of a nanofluid past a vertical plate Int. J. Thermal Sci.49 243 247 - 45.
Kuznetsov AV, Nield DA 2010 Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid Transp. Porous Media83 425 436 - 46.
Bachok N. Ishak A. Pop I. 2010 Boundary-layer flow of nanofluids over a moving surface in a flowing fluid Int. J. Thermal Sci.49 1663 1668